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EE370/01 Examination No. 1 Fall 2008/09
(50 minutes)
Kuwait University
Electrical Engineering Department
Name :
Student I. D. : .
Signature : .
Problem No. Grade
1 25
2 25
3 25
4 25
Total 100
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Problem 1 (25 points): Drive the expression for)(
)()(
sR
sUsM in terms of )(sGi for the following
system:
Problem 2 (25 points): Can you approximate the step response)10()(
)5.10()(
2
sbsass
ssC as
the step response of 2nd order system with5
2damping ratio and 2 seconds settling time? If yes,
find the values of a and b? Justify your answer.
Problem 3 (25 points): Use the Routh Table to find the range of K that keeps the following
feedback system stable.
Problem 4 (25 points): (a) Find the steady state error, )( te , in terms of a and b for the
following system:
(b) Find the values of a and b that yield zero steady state error.
+
+
_ _
C(s)R(s)
+
_
++
_
++
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EE370/01 Examination No. 1 Fall 2006/07
Kuwait University
Electrical Engineering Department
Name :
Student I. D. : .
Signature : .
Problem No. Grade
1 20
2 20
3 20
4 20
5 20
Total 100
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Problem 1 (20 points): Can you use the partial fraction expansion2
43
2
2
2
1
)2(211
s
k
s
k
s
sk
s
k
to find the inverse Laplace transform of22
2
)2)(1(
3
ss
sif:
a) 01
k b) 0
3 k
c) 32 kk
Explain your answer.
324
044
124
0
)24()44()24()(
)1()2)(1()2()2(3
431
321
4321
32
431321
2
4321
3
32
2
4
2
3
2
2
2
1
2
kkk
kkk
kkkk
kk
kkkskkkskkkkskk
sksskssksks
a) No, you cannot solve the equationsb) No, you cannot solve the equationsc) Yes, you can solve the equations.
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Problem 2 (20 points):
a) (10 points): Drive the expression of the output )(sC in terms of( ), ( ), ( ), ( ), ( )M s H s G s R s and D s for the following system:
b)
(10 points): What conditions on ( )M s and )(sG you will impose if you want to make thesteady state output )(tc equals zero when 0)( sR and )(sD is a step function.
LHPtheinGMofrootstheallplus
MdMG
HG
s
d
GM
HGsCb
DGM
HGR
GM
GCHDMCRHDG
HDGXCa
sss
1
)0(0)0()0(1
)0())0(1(
1
)1(lim)
1
)1(
1
)(
)
+
+
+
+
-
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Problem 3 (20 points): Consider the following closed-loop system:
Find the values of1
k and2
k to get 0.5 settling time.
ionapproximatorderndensuretok
kkw
T
kw
kksks
ks
ss
ksk
ss
ks
sT
n
s
n
20
145.02
84
22
)10()2(
1021
102)(
1
2
2
2
212
2
1
2
1
2
2
1
C(s)R(s)
+
_
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Problem 4 (20 points): Use Routh-Hurwitz criterion to find the values of0
c ,1
c ,2
c that will cause
the polynomial01
2
2
34)( cscscsssM to have two real roots in the LHP, one real root in
the RHP, and one root on the j-axis.
4s 1 2c 0c
3s 1 1c 2
s 12 cc 0c
1s
12
01121
cc
ccccc
0s 0c
Make rows 2 equal zero:12
cc and 00 c
Form the polynomial from scsrows 133
:
Differentiate the polynomial:1
23 cs
You have 3 symmetrical roots, select 01
c (one sign change): one root on the RHP and its
symmetrical root on the LHP plus one root on jw-axis. Also, one root on the LHP since you have
4 roots in total.
Then 012 cc and 00 c
4s 1 2c 0c
3s 1 1c 2
s 3 1c 1
s 1
11
3
2
3
3c
cc
0s 1c
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Problem 5 (20 points): A unity negative feedback system has the open loop transfer function
2
1)(
2
21
ss
skksG . Determine the gains
1k and
2k that minimize the steady state error
due to the step, ramp, or parabolic input.
)(2
)2(lim)(
2
11
lim)()(1
lim)(12
23
3
0
2
21
00 sRkskss
sssR
ss
skk
ssR
sG
ste sss
Then 05.0 12 kk for stability3
s 1 2k 2
s 2 1k 1s 12 5.0 kk 0
s 1k
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EE370/01 Examination no. 1 Fall 2005/06
Kuwait University
Electrical Engineering Department
Name :
Student I.D. :.
Problem no. Grade1 20
2 20
3 20
4 20
5 20
Bonus 20
Total 120
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Problem 1 (20 points): Given the following differential equation:
2
2
( ) ( ) ( 0.01)3 2 ( ) ( 0.01).
d y t d y t d u t y t u t
dt dt dt
Use Laplace transform to find ( )y t for a unit step input ( )u t and zero initial conditions.
Problem 2 (20 points): Find the value of 0a that will cause the following system to have somepoles on the j-axis. Use Routh-Hurwitz criterion.
Problem 3 (20 points):
c) (10 points): Drive the expression of the error ( ) ( ) ( )E s R s C s in terms of( ), ( ), ( ), ( ), ( )M s H s G s R s and D s for the following stable system:
d) (10 points): If ( )R s and ( )D s are unit step functions, find the relation between( )H s and ( )M s in order to get zero steady state error.
+
-
+
- + +
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Problem 4 (20 points): Consider the following closed-loop system:
Can you find the values of 1 0k and 2 0k to meet the following specifications:
Closed-loop damping ratio 0.8 . Closed-loop settling time 10sT sec. Steady state error of 0.1 due to a unit step input.
Write the reason if you cannot find 1k and 2k .
Problem 5 (20 points):a) (16 points): Sketch the root locus for the following system.
Show the following on the root locus:
Breakaway and/or break-in points. Angles of departure and/or arrival.
b) (4 points): Can you find 0k to stabilize the closed-loop system.
Bonus Problem (20 points): Consider the following feedback system
where 7 1a . Find the value of 0k that will minimize the steady state error due to a unitstep input for
C(s)R(s)+ +
- -
+
-
+
-
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EE370/01 Examination no. 1 Spring 2006
Kuwait University
Electrical Engineering Department
Name :
Student I.D. :.
Problem no. Grade
1 25
2 25
3 25
4 25
Bonus 20
Total 120
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Problem 1 (25 points): Given the closed-loop transfer function2
( ) 3 4( ) :
( ) 12 12
Y s sT s
R s s s
for the
following system
e) (10 points): Determine the closed-loop poles and zeros. Also, show if there is any dominantpole(s).
f) (15 points): Find the transfer function ( )G s ?
Problem 2 (25 points): Given the following system
a) (10
points): Determine the value of 0k that will minimize the steady state error
due to a unit step input ( )r t .
b) (15 points): Determine the value of 0k that will minimize the sensitivity of the error ( )E s to a change in the parameter a .
Problem 3 (25 points): Given3 2
( ) 6 M s s s ks c , find the range of k in term of c so that all
the roots of ( )M s have a negative real part. Use Routh-Hurwitz criterion.
Problem 4 (25 points): Consider the system:
Determine the values of , ,a b c and k to have a unit step response ( )y t with: steady state value of 2,
maximum amplitude of 3 and settling time of 4 sec.
+
-
2
-
k
-
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Bonus (20 points): Given the Laplace transform of
( ( 1))d f a t
dt
equals G(s), find the
Laplace transform of ( )f t in term of G(s)?
EE370/51 Examination No. 1 Spring 2009
(50 minutes)
Kuwait University
Electrical Engineering Department
Name (Arabic) :
Student I. D. : .
Signature : .
Problem No. Grade
1 25
2 25
3 25
4 25
Total 100
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Problem 1 (25 points): For the following system, find )(sU ?
++
+_ __
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Problem 2 (25 points): Consider the following translational mechanical system. A 2 N force f(t)
is applied for 0t , find the values of vf , M, and K such that the response x(t) is given by the
plot shown below.
0 0.5 20
1
1.1
Response
x(t)
Step Response
Time (sec)
Amplitude
fv
K
f (t)
x (t)
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Problem 3 (25 points): Use the Routh Table to find the range of that makes the followingfeedback system unstable.
Problem 4 (25 points): (a: 15 points) Derive the expression for the error )()()( sCsRsE in term
of )(sH for the following system:
(b: 10 points) Find the value of )0( sH that minimizes the steady state error.
C(s)R(s)+
_
+
_
+
+